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Partial Functions in ACL2
 Journal of Automated Reasoning
"... We describe a macro for introducing \partial functions" into ACL2, i.e., functions not dened everywhere. The function \denitions" are actually admitted via the encapsulation principle. We discuss the basic issues surrounding partial functions in ACL2 and illustrate theorems that can be proved ab ..."
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We describe a macro for introducing \partial functions" into ACL2, i.e., functions not dened everywhere. The function \denitions" are actually admitted via the encapsulation principle. We discuss the basic issues surrounding partial functions in ACL2 and illustrate theorems that can be proved about such functions.
On the nonexistence of 2cycles for the 3x + 1 problem
 Mathematics of Computation
"... Abstract. This article generalizes a proof of Steiner for the nonexistence of 1cycles for the 3x + 1 problem to a proof for the nonexistence of 2cycles. A lower bound for the cycle length is derived by approximating the ratio between numbers in a cycle. An upper bound is found by applying a result ..."
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Cited by 3 (2 self)
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Abstract. This article generalizes a proof of Steiner for the nonexistence of 1cycles for the 3x + 1 problem to a proof for the nonexistence of 2cycles. A lower bound for the cycle length is derived by approximating the ratio between numbers in a cycle. An upper bound is found by applying a result of Laurent, Mignotte, and Nesterenko on linear forms in logarithms. Finally numerical calculation of convergents of log2 3 shows that 2cycles cannot exist. 1.
Lagarias, Lower bounds for the total stopping time of 3x +1 iterates
 Math. Comp
"... Abstract. The total stopping time σ∞(n) of a positive integer n is the minimal number of iterates of the 3x + 1 function needed to reach the value 1, and is + ∞ if no iterate of n reaches 1. It is shown that there are infinitely many positive integers n having a finite total stopping time σ∞(n) such ..."
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Abstract. The total stopping time σ∞(n) of a positive integer n is the minimal number of iterates of the 3x + 1 function needed to reach the value 1, and is + ∞ if no iterate of n reaches 1. It is shown that there are infinitely many positive integers n having a finite total stopping time σ∞(n) such that σ∞(n)> 6.14316 log n. The proof involves a search of 3x + 1 trees to depth 60, A heuristic argument suggests that for any constant γ<γBP ≈ 41.677647, a search of all 3x + 1 trees to sufficient depth could produce a proof that there are infinitely many n such that σ∞(n)> γlog n. It would require a very large computation to search 3x + 1 trees to a sufficient depth to produce a proof that the expected behavior of a “random ” 3x + 1 iterate, which is γ = 2 ≈ 6.95212, occurs infinitely often. log 4/3 1.
Theoretical and Computational Bounds for MCycles of the 3n + 1 Problem
, 2004
"... An mcycle of the 3n+1problem is defined as a periodic orbit with m local minima. In this article we derive lower and upper bounds for the cycle length and the elements of (hypothetical) mcycles. ..."
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An mcycle of the 3n+1problem is defined as a periodic orbit with m local minima. In this article we derive lower and upper bounds for the cycle length and the elements of (hypothetical) mcycles.
A Binomial Representation of the 3x + 1 problem
"... We show how the 3x + 1 conjecture can be expressed in the language of arithmetical formulas with binomial coecients. 1 ..."
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We show how the 3x + 1 conjecture can be expressed in the language of arithmetical formulas with binomial coecients. 1
Natural orbital networks
"... Abstract. These are some informal remarks on quadratic orbital networks over finite fields Zp. We discuss connectivity, Euler characteristic, number of cliques, planarity, diameter and inductive dimension. We prove that for d = 1 generators, the Euler characteristic is always nonnegative and for d = ..."
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Abstract. These are some informal remarks on quadratic orbital networks over finite fields Zp. We discuss connectivity, Euler characteristic, number of cliques, planarity, diameter and inductive dimension. We prove that for d = 1 generators, the Euler characteristic is always nonnegative and for d = 2 and large enough p the Euler characteristic is negative. While for d = 1, all networks are planar, we suspect that for d ≥ 2 and large enough p, all networks are nonplanar. As a consequence on bounds for the number of complete subgraphs of a fixed dimension, the inductive dimension of all these networks goes 1 as p → ∞. [December 23 update: longer runs] 1. Polynomial orbital networks Given a field R = Zp, we study orbital graphs G = (V, E) defined by polynomials Ti which generate a monoid T acting on R. We think of (R, T) as a dynamical system where positive time T is given by the monoid of words w = w1w2... wk using the generators wk ∈ A = {T1,..., Td} as alphabet and where {T w x  w ∈ R} is the orbit of x. The orbital network [1, 3, 4] is the finite simple graph G where V = R is the set of vertices and where two vertices x, y ∈ V are connected if there exists Ti such that Ti(x) = y or Ti(y) = x. The network generated by the system consists of the union of all orbits. As custom in dynamics, one is interested in invariant components of the system and especially forward attractors Ω(x) of a point x as well as the garden of eden, the set of points which are not in the image of any Ti. We are also interested in the inductive dimension of the network. This relates to the existence and number of cliques, which are complete subgraphs of G.
LOWER BOUNDS FOR ZNUMBERS
"... Abstract. Let p/q be a rational noninteger number with p> q ≥ 2. A real number λ> 0 is a Z p/qnumber if {λ(p/q) n} < 1/q for every nonnegative integer n, where {x} denotes the fractional part of x. We develop several algorithms to search for Z p/qnumbers, and use them to determine lower b ..."
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Abstract. Let p/q be a rational noninteger number with p> q ≥ 2. A real number λ> 0 is a Z p/qnumber if {λ(p/q) n} < 1/q for every nonnegative integer n, where {x} denotes the fractional part of x. We develop several algorithms to search for Z p/qnumbers, and use them to determine lower bounds on such numbers for several p and q. It is shown, for instance, that if there is a Z 3/2number then it is greater than 2 57. We also explore some connections between these problems and some questions regarding iterated maps on integers. 1. An approximate multiplication problem Let us begin with the following problem. Starting with a positive integer x, we consider the map ⎪ ⎨ 4x/3, if x ≡ 0 (mod 3),
T: Z → Z, n ↦→
"... ABSTRACT. Motivated by the famous 3n + 1 Conjecture, we call a mapping from Z to Z residueclasswise affine if there is a positive integer m such that it is affine on residue classes (mod m). This article describes a collection of algorithms and methods for computing in permutation groups and monoi ..."
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ABSTRACT. Motivated by the famous 3n + 1 Conjecture, we call a mapping from Z to Z residueclasswise affine if there is a positive integer m such that it is affine on residue classes (mod m). This article describes a collection of algorithms and methods for computing in permutation groups and monoids formed by residueclasswise affine mappings.
The Generalized 3x + 1 Mapping
, 2000
"... Introduction One of the most tantalizing conjectures in number theory is the socalled 3x + 1 conjecture, attributed to L. Collatz (1937): Let T : Z # Z be defined by T (x) = # # # x/2 if x # 0 (mod 2) (3x + 1)/2 if x # 1 (mod 2). (1) Collatz conjectured that if x # N, then the trajector ..."
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Introduction One of the most tantalizing conjectures in number theory is the socalled 3x + 1 conjecture, attributed to L. Collatz (1937): Let T : Z # Z be defined by T (x) = # # # x/2 if x # 0 (mod 2) (3x + 1)/2 if x # 1 (mod 2). (1) Collatz conjectured that if x # N, then the trajectory x, T (x), T 2 (x), . . . eventually reaches the cycle 1, 2, 1. If x # N, it is also conjectured that the trajectory {T k (x)} eventually reaches one of the cycles (a) 1, 1; (b)
THE SUFFICIENCY OF ARITHMETIC PROGRESSIONS FOR THE 3x +1 CONJECTURE
"... Abstract. Define T: Z + → Z + by T (x) = (3x+1)/2 if x is odd and T (x) =x/2 ifxis even. The 3x+1 Conjecture states that the Torbit of every positive integer contains 1. A set of positive integers is said to be sufficient if the Torbit of every positive integer intersects the Torbit of an elemen ..."
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Abstract. Define T: Z + → Z + by T (x) = (3x+1)/2 if x is odd and T (x) =x/2 ifxis even. The 3x+1 Conjecture states that the Torbit of every positive integer contains 1. A set of positive integers is said to be sufficient if the Torbit of every positive integer intersects the Torbit of an element of that set. Thus to prove the 3x+1 Conjecture it suffices to prove it on some sufficient set. Andaloro proved that the sets 1 + 2nN are sufficient for n ≤ 4andasked if 1 + 2nN is also sufficient for larger values of n. We answer this question in the affirmative by proving the stronger result that A + BN is sufficient for any nonnegative integers A and B with B ̸ = 0, i.e. every nonconstant arithmetic sequence forms a sufficient set. We then prove analagous results for the Divergent Orbits Conjecture and Nontrivial Cycles Conjecture.